Integrated sensing and communication (ISAC) poses various challenges that arise from the communication-centric design of cellular networks. One of them is target detection with time division duplex (TDD) transmission used in current 5G and future 6G deployments, where the periodic on-off behavior of the transmitter creates impulsive sidelobes in the radar point spread function (PSF). These can be mistaken for actual targets by conventional peak detection techniques, leading to false alarms. In this work, we first analytically describe the range-Doppler PSF due to TDD windowing. We then propose a computationally efficient method that leverages the PSF to distinguish impulsive sidelobes from valid target peaks. Simulation results and outdoor drone measurements with an ISAC proof of concept demonstrate the capability of our algorithm, showing that it can achieve reliable target detection while limiting false alarms.